The weak Lefschetz property of Gorenstein algebras of codimension three associated to the Apery sets

It has been conjectured that allgraded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras A of the Apery set of M-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if A is not a complete intersection, then A is of form A = R/I with R = K[x, y, z] and I = (x(a), y(b) - x(b-gamma)z(gamma),z(c),x(a-b+gamma)y(b-beta),y(b-beta)z(c-gamma)), where 1 <= beta <= b - 1, max{1, b - a + 1} <= gamma <= min{b - 1, c - 1} and a >= c >= 2. We prove that Ahas the weak Lefschetz property in the following cases: max{1, b - a + c - 1} <= beta <= b - 1 and gamma >= left perpendicular beta-a+b+c-2 right perpendicular; a <= 2b - cand vertical bar a - b vertical bar + c - 1 <= beta <= b - 1; one of a, b, c is at most five (C) 2020 Elsevier Inc. All rights reserved.